We study zero-error entanglement assisted source-channel coding (communication in the presence of side information). Adapting a technique of Beigi, we show that such coding requires existence of a set of vectors satisfying orthogonality conditions related to suitably defined graphs G and H. Such vectors exist if and only if theta(G) <= theta(H) where theta represents the Lovász number. We also obtain similar inequalities for the related Schrijver theta^- and Szegedy theta^+ numbers. These inequalities reproduce several known bounds and also lead to new results. We provide a lower bound on the entanglement assisted cost rate. We show that the entanglement assisted independence number is bounded by the Schrijver number: alpha^*(G) <= theta^-(G). Therefore, we are able to disprove the conjecture that the one-shot entanglement-assisted zero-error capacity is equal to the integer part of the Lovász number. Beigi introduced a quantity beta as an upper bound on alpha^* and posed the question of whether beta(G) = \lfloor theta(G) \rfloor. We answer this in the affirmative and show that a related quantity is equal to \lceil theta(G) \rceil. We show that a quantity chi_{vect}(G) recently introduced in the context of Tsirelson's conjecture is equal to \lceil theta^+(G) \rceil.
@InProceedings{cubitt_et_al:LIPIcs.TQC.2014.48, author = {Cubitt, Toby and Mancinska, Laura and Roberson, David and Severini, Simone and Stahlke, Dan and Winter, Andreas}, title = {{Bounds on Entanglement Assisted Source-channel Coding Via the Lov\'{a}sz Theta Number and Its Variants}}, booktitle = {9th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2014)}, pages = {48--51}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-73-6}, ISSN = {1868-8969}, year = {2014}, volume = {27}, editor = {Flammia, Steven T. and Harrow, Aram W.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.TQC.2014.48}, URN = {urn:nbn:de:0030-drops-48054}, doi = {10.4230/LIPIcs.TQC.2014.48}, annote = {Keywords: source-channel coding, zero-error capacity, Lov\'{a}sz theta} }
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